EXAMPLE 14Recognizing binomial experiments

Determine whether each of the following experiments fulfills the conditions for a binomial experiment. If the experiment is binomial, identify the random variable , the number of trials, the probability of success, and the probability of failure. If the experiment is not binomial, explain why not.

  1. A fisherman is going fishing and will continue to fish until he catches a rainbow trout.
  2. We flip a fair coin three times and observe the number of heads.
  3. A market researcher at a shopping mall is asking consumers whether they use Fib detergent. She asks a sample of four men, one of whom is clearly the employer of the other three.
  4. The National Burglar and Fire Alarm Association reports that 34% of burglars get in through the front door. A random sample of 36 burglaries is taken, and the number of entries through the front door is noted.

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Solution

  1. This is not a binomial experiment because you don't know how many fish he will catch before the rainbow trout shows up, so a fixed number of trials isn't known in advance.

  2. This is a binomial experiment because it fulfills the requirements:

    1. Only two possible outcomes are possible on each trial, with heads defined as success and tails as failure.
    2. We know in advance that we are tossing the coin three times.
    3. The coin doesn't remember its result from toss to toss, and so the trials are independent.
    4. The coin is fair on each toss, and so the probability of observing heads is the same on each toss.

    The binomial random variable is the number of heads observed on the three trials; because the coin is fair, the probability of success is 0.5 and the probability of failure is 0.5. The possible values for are 0, 1, 2, or 3.

  3. This is not a binomial experiment because the responses are not independent. The response given by the employer is likely to affect the employees' responses.

  4. This is a binomial experiment because it fulfills the requirements:

    1. Only two possible outcomes are possible on each trial: entering through the front door or not entering through the front door.
    2. We know in advance that the size of the random sample is 36 burglaries.
    3. The sample is random, so the trials are independent.
    4. The sample is quite small compared to the size of the population, so the probability of entering through the front door remains the same from burglary to burglary.

The binomial random variable is the number of front-door-entry burglaries noted for the 36 break-ins; the probability of success is 0.34 and the probability of failure is .

NOW YOU CAN DO

Exercises 5–14.